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    Wyświetlanie 1-3 z 3
    Tytuł:
    Gödel’s Philosophical Challenge (to Turing)
    Autorzy:
    Sieg, Wilfried
    Powiązania:
    https://bibliotekanauki.pl/articles/1796958.pdf
    Data publikacji:
    2020
    Wydawca:
    Polskie Towarzystwo Semiotyczne
    Tematy:
    computability
    Church's Thesis
    Turing's Thesis
    incompleteness
    undecid-ability
    Post production systems
    computable dynamical systems
    Opis:
    The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that non-mechanical steps of intuition are needed to transcend particular formal theories. Thus, there is a substantive point in comparing Turing’s views with Gödel’s that is expressed by the assertion, “The human mind infinitely surpasses any finite machine”. The parallelisms and tensions between their views are taken as an inspiration for beginning to explore, computationally, the capacities of the human mathematical mind.
    Źródło:
    Studia Semiotyczne; 2020, 34, 1; 57-80
    0137-6608
    Pojawia się w:
    Studia Semiotyczne
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Gödel’s Incompleteness Theorem and the Anti-Mechanist Argument: Revisited
    Autorzy:
    Cheng, Yong
    Powiązania:
    https://bibliotekanauki.pl/articles/1796969.pdf
    Data publikacji:
    2020
    Wydawca:
    Polskie Towarzystwo Semiotyczne
    Tematy:
    Gödel’s incompleteness theorem
    the Anti-Mechanist Argument
    Gödel’s Disjunctive Thesis
    intensionality
    Opis:
    This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper (2020). This paper gives some supplementary notes to Krajewski’s (2020) on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s (2020) and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 7–8 in Krajewski’s (2020), describing how recent advances bear on these issues.
    Źródło:
    Studia Semiotyczne; 2020, 34, 1; 159-182
    0137-6608
    Pojawia się w:
    Studia Semiotyczne
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Anti-Mechanist Argument Based on Gödel’s Incompleteness Theorems, Indescribability of the Concept of Natural Number and Deviant Encodings
    Autorzy:
    Quinon, Paula
    Powiązania:
    https://bibliotekanauki.pl/articles/1796973.pdf
    Data publikacji:
    2020
    Wydawca:
    Polskie Towarzystwo Semiotyczne
    Tematy:
    the Lucas-Penrose argument
    the Church-Turing thesis
    Carnapian expli-cations
    natural numbers
    computation
    conceptual engineering
    conceptual fixed points
    conceptual vicious circles
    deviant encodings
    structuralism
    Opis:
    This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski (2020): this argument only works with the additional extra-formal assumption that “the human mind is consistent”. Krajewski argues that this assumption cannot be formalized, and therefore that the anti-mechanist argument – which requires the formalization of the whole reasoning process – fails to establish that the human mind is not mechanistic. A similar situation occurs with a corollary to the argument, that the human mind allegedly outperforms machines, because although there is no exhaustive formal definition of natural numbers, mathematicians can successfully work with natural numbers. Again, the corollary requires an extra-formal assumption: “PA is complete” or “the set of all natural numbers exists”. I agree that extra-formal assumptions are necessary in order to validate the anti-mechanist argument and its corollary, and that those assumptions are problematic. However, I argue that formalization is possible and the problem is instead the circularity of reasoning that they cause. The human mind does not prove its own consistency, and outperforms the machine, simply by making the assumption “I am consistent”. Starting from the analysis of circularity, I propose a way of thinking about the interplay between informal and formal in mathematics.
    Źródło:
    Studia Semiotyczne; 2020, 34, 1; 243-266
    0137-6608
    Pojawia się w:
    Studia Semiotyczne
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-3 z 3

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